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May 1, 2026Open Access

A Geometric Identity for the Fine-Structure Constant: From the 4D Unit Ball Volume and its Cube-Packing Deficit (Paper 7, v3)

Key Points

This work aims to establish a geometric identity for the fine-structure constant α using 4-dimensional quantities.
Derivation of the identity α⁻¹ = N(1) + V_4(1)·α from 4D geometric properties.
Calculation of N(1) = 137 and V_4(1) = π²/2 to evaluate the quadratic equation.
High-precision verification for accuracy against CODATA values.
The derived value of α = 7.29735194×10⁻³ aligns with the CODATA 2018 measurement at 8.7 ppb accuracy.
The identity reveals differences from Eddington-style integer-fitting by being self-consistent and based on geometric interpretation.
The quadratic formulation offers insights into the relationship between geometric constructs and quantum field theory.

Abstract

This paper presents a self-consistent algebraic identity for the fine-structure constant α from two elementary 4-dimensional geometric quantities: (i) N (1) = 137, the count of integer-centred unit cubes fully contained in a 4D ball of radius 3 (re-derived by direct enumeration), and (ii) V₄ (1) = π²/2, the volume of the 4D unit ball. The identity α⁻¹ = N (1) + V₄ (1) ·α is equivalent to a self-consistent quadratic (π²/2) α² + 137α − 1 = 0 whose positive root predicts α = 7. 29735194×10⁻³, agreeing with the CODATA 2018 value 7. 29735257×10⁻³ to a relative accuracy of 8. 7×10⁻⁸ (about 8. 7 ppb) — approximately 1/3000 the deviation of Eddington-style integer-fitting. The identity admits a perturbative reading 1 = 137α + V₄ (1) α², interpreted as "tree-level coupling on the inside (137 packed cubes) plus self-energy correction on the outside (boundary gap, 66% of the R=3 ball volume) ". Decisive differences from Eddington-style integer-fitting are spelled out: self-consistent (not pure integer) ; both 137 and π²/2 derived independently from 4D geometry; QFT-perturbative analogue. §6. 5 addresses the dimensional mismatch via Schläfli duality between the 4D hypercube and 16-cell (137 cubes ↔ 137 nodes on the dual lattice), establishing structural correspondence with QED Feynman vertex rules and affinity with spin networks (LQG). v3 changes (2026-04-29): Numerical precision correction. Previous v1/v2 reported "0. 02% accuracy" based on coarse 4-digit rounding (α ≈ 7. 2984×10⁻³). High-precision external verification (Grok / xAI, mpmath 50-digit) showed the actual relative error is 8. 7×10⁻⁸ (8. 7 ppb), three orders of magnitude better than originally claimed. Abstract, §3. 2 numerical table, §6. 1 residual analysis, §7 conclusion, and Appendix updated. The 0. 03% small-scale residual interpretation is replaced with the corrected 8. 7 ppb value. Whether the identity reflects geometric necessity or numerical coincidence is left as an open problem. The 8. 7 ppb residual and first-principles derivation of the correction term are flagged as principal open questions. Companion paper to the BH Thermodynamics Programme (Papers 1–6).

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